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LESSON OBJECTIVES 

9.1 Patterns 

• Identify and write a 

formula for a pattern. 

• Use a formula to find the 

terms in a sequence. 

9.2 Arithmetic Sequences 

and Series 

• Identify an arithmetic 

sequence. 

• Write and evaluate an 

arithmetic series. 

9.3 Geometric Sequences 

and Series 

• Identify a geometric 

sequence. 

• Write and evaluate a 

geometric series. 

9.4 Infinite Geometric Series 

• Find the sum of an infinite 

geometric series. 

• Determine the difference 

between divergence and 

convergence. 

9.5 The Binomial Theorem 

• Use Pascal’s Triangle and 

the Binomial Theorem 

to expand powers of a 

binomial. 

Mental Math 

Find each sum. 

1. 2 + 6 + 10 + 14 + 18 50 

2. 1.5 + 3.5 + 5.5 + 7.5 + 9.5 

27.5 

3. 3 + 5.5 + 8 + 10.5 + 13 40 

4. 1.25 + 1.9 + 2.55 + 3.2 

+ 3.85 12.75 

 

Simplify. 

1. 1− 2 1 

3 

1 

3 

1 

2. 1− 4 5 

5 

 

 

 

 

 

 

 

 

 

Skills Review 

 

 

 

 

 

 

3. 1 − 1 4 27 

4 16 

9 

1 

4. 1+ 7 10 

10 17 

390 Chapter 9 Sequences and Series

Look to Your Future 

Computer programmers use programming languages to write sequences. 

When followed in the correct order, these sequences cause a particular action 

to take place. This sequence or line of instruction is the computer program. 

Some computer programs can be written in hours, while other programs that 

use complex mathematical formulas, can take a year or more to create. 

PLANNING 

THE CHAPTER 

Math Labs, pp. 420–421 

Data Sheet (Lab Data Sheets) 

Math Applications, pp. 422–429 

Chapter Review, pp. 430–431 

Chapter Test, p. 432 

Software Generated Assessment 

Standardized Test Practice, p. 433 

Grid Response Form (CRB) 

Chapter Resource Book (CRB) 

Reteaching, pp. 305, 309, 315, 

319, 325 

Extra Practice, pp. 307, 311, 

317, 321, 327 

Enrichment, pp. 313, 323 

Standardized Test Response 

Form, pp. 329, 330 

Standardized Test Answers, 

p. 331 

Classroom/Journal 

Topics 

What’s Ahead? 

In this chapter, students will 

learn about patterns, sequences, 

and series. Students will use 

Pascal’s Triangle and the Binomial 

Theorem to expand powers of a 

binomial. 

Many real-life situations are given 

where geometric sequences are 

written and evaluated. Students 

should be able to differentiate 

between arithmetic and 

geometric sequences. 

Chapter 9 Sequences and Series 391

LESSON PLANNING 

Vocabulary 

sequence 

term 

recursive formula 

Fibonacci sequence 

explicit formulas 

Extra Resources 

Reteaching 9.1 

Extra Practice 9.1 

Assignment 

In-class practice: 1–5 

Homework: 6–38 

Math Applications 

Exercises 3, 6, 7, and 15 from 

pages 422–429 

START UP 

Using a spreadsheet, ask students 

to help you make a table of values 

which represents the amount 

of money in Maryanne’s savings 

account. Guide students to 

describe two different ways that 

can be done. 

ACTIVE LEARNING 

Activity 

Challenge students to find a 

recursive formula for the second 

column of the table. 

Modify the table so that it shows 

the volumes of cubes with sides 

from 1 to 5 units in length. Ask 

students to complete questions 

1–4 again. 

 

 

 

 

 

 

 

 

R.E.A.C.T. Strategy 

Cooperating 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Have each student create a sequence of 5 terms using an explicit rule. Have 

students exchange papers and answer questions 1–4 used in the activity about 

the sequence created by their partner. As a further challenge, ask students to 

find a recursive formula for the sequence. 


INSTRUCTION 

Recursion is a new topic for 

students. Tell students that 

recursion is demonstrated in the 

following joke: Pete and Repeat 

sat on a fence. Pete fell off. 

Who was left? Repeat. Pete and 

Repeat . . . 

Tell students that recursion can 

also be seen where two mirrors 

are placed almost parallel to each 

other. The image in a mirror is an 

infinite sequence of reflections, 

each a little smaller version than 

the previous, disappearing into 

the distance. 

Ongoing Assessment 

Show students the explicit 

formula for this problem also as 

a reminder of the exponential 

functions which were studied in 

Chapter 6. 

 

 

 

 

 

 

 

Diversity in the Classroom 

Visual Learner 

Direct students to an Internet site which will allow them to interactively play 

the classic puzzle “The Towers of Hanoi.” Instruct students to begin with one 

disc, and continue on through five discs. Tell students to record the minimum 

number of moves that are required for each number of discs. Ask students 

to identify the resulting sequence and to describe the recursive nature of the 

game. 

9.1 Patterns 393

INSTRUCTION 

Example 2 Return to 

the spreadsheet used at the 

beginning of class. Remind 

students of the formulas that 

they created to represent the 

amount of savings in Maryanne’s 

account. Ask students to classify 

the formulas as being recursive or 

explicit. 

Ask students to identify which 

method of defining a sequence 

they prefer if given a choice. Ask 

students to justify their answers. 

Think and Discuss 

Answers 

1. No, the value of a 1 

is not 

known. 

2. Answers will vary. Sample 

answer: a sequence is an 

ordered list of numbers such 

as 2, 4, 6, 8, 10, … 

3. An explicit formula is given by 

a rule that does not depend 

on other terms of the pattern. 

A recursive formula generates 

terms of the pattern based on 

previous terms. 

4. The formula is explicit since 

a n 

does not depend on a 

previous term. 

5. Substitute 8 for n in the 

expression and simplify. 

 

 

 

 

 

 

 

 

 

 

 

 


Applying 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Introduce students to the classic handshake problem. Challenge students to 

determine how many handshakes would take place in a roomful of 20 people. 

Ask students to justify their answers and to identify if they used an explicit or 

recursive formula when solving the problem. 


WRAP UP 

To ensure mastery of objectives, 

students should be able to: 

• Write the appropriate recursive 

or explicit formula for a given 

sequence. 

• Use an explicit or recursive 

formula to determine the 

terms in a sequence. 

Reteaching 9.1 (CRB) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Extra Practice 9.1 (CRB) 

 

Assignment 




Exercises 3, 6, 7, and 15 from pages 422–429 

9.1 Patterns 395

Answers to Math 

Applications 

Math Applications for this 

chapter are on pages 422–429. 

The notes and solutions shown 

below accompany the suggested 

applications to assign with this 

lesson. 

3. a. 125 + 0.4x 

b. 125 + 0.4(300) = 125 + 

120 = 245 

c. 25 + 0.6x 

d. 25 + 0.6(200) 

= 25 + 120 = 145 

6. a. a 2 

= $750,000(1.1) 

= $825,000 

b. a 3 

= $825,000(1.1) 

= $907,500 

$905,000 < $907,500, so 

Mari did not earn a bonus. 

c. a 5 

= $750,000(1.1) 5–1 = 

$750,000(1.4641) 

= 1,098,075 

d. yes; a 2 

would now equal 

$900,000 instead of 

$825,000 and would 

increase each subsequent 

number. 

e. a n 

= 1.1(a n–1 

) 

7. a. For each 1 cent change 

in earnings per share, the 

closing share price changes 

0.68. The expression is 

0.68x. 

b. 0.68(9) = $6.12 

15. a. Each term is the product of 

the previous two terms, so 

a n 

= a n–2 

• a n–1 

b. 0.3 = 0.5(a n–2 

) 

0.6 = a n–2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 

24 

1 

16 

1 

36 

1 

40 

 

 

 

 

 

 

 

 

2 de 

3 7 

f 

 

 

 

 

 

 

 



Vocabulary 

arithmetic sequence 

common difference 

arithmetic mean 

series 

summation notation 




Enrichment 9.2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer to Ongoing Assessment 

 

Yes, the pattern is an arithmetic sequence with a common difference of 0.6. 

The next 3 terms of the sequence are 4.0, 4.6, and 5.2. 

Assignment 





pages 422–429 

START UP 

Write the following sequences on 

the board. 

a) 1, 3, 5, 7, 9, 11 . . . 

b) 4, 9, 16, 25, 36 . . . 

c) 12, 9, 6, 3, . . . 

d) 3, 9, 27, 81, 243, . . . 

Tell students that sequences a and 

d are called arithmetic sequences, 

but that sequences b and c are 

not. Ask students to deduce 

the definition of an arithmetic 

sequence. 

INSTRUCTION 

Tell students that an arithmetic 

sequence can be defined both 

using a recursive formula and 

using an explicit formula. 

Example 1 Ask students if 

this sequence implies that the 

distance between bus stops is the 

same. (no) 

9.2 Arithmetic Sequences and Series 397

INSTRUCTION 

Tell students that the arithmetic 

mean of two numbers is the 

number that is equally distant 

between the two numbers in 

addition. 

Point out that since there is a 

common difference between 

consecutive terms in an arithmetic 

sequence, that the difference 

between every other term is also a 

constant. The difference between 

every other term is twice the 

common difference. 

Answer to Critical 

Thinking 

Answers will vary. Sample answer: 

Let d represent the common 

difference of the sequence. Then 

the first missing term is 11 + d, 

and the second missing term is 

11 + d + d, or 11 + 2d, and 

the next term, 32, is equal to 

11 + 2d + d, or 11 + 3d. Solve 

the equation 11 + 3d = 32 for d 

to find the missing terms: 18 and 

25. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

105 135 

2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Enriching the Lesson 

Show students the following alternative way to determine 

the missing term in Example 2. 

Step 1: The difference between the terms 105 and 135 is 30. 

Step 2: Because the difference between every other term in an arithmetic 

sequence is twice the common difference, the common difference is 

30÷2 = 15. 

Step 3: Adding the common difference to 105 yields a missing term of 120. 


5 

∑ 

= 1 

 

3 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 

∑ 

= 

1 

12 + 12 

 

 

 

 

 

 

 

20 2 

 


 

Tell students the following legend: When the famous mathematician 

Leonhard Euler was in grade school, the school master, in an 

attempt to keep his students busy, told the students to find the sum of the 

natural numbers from 1 to 100. The school master was quite disappointed 

when Leonhard was able to produce the correct answer in about sixty seconds. 

Show students the following method that was used by Euler. Then show 

students how the formula S = 

n 

( a + a ) is a generalization of that method. 

n 

2 1 

1 2 3 . . . . 98 99 100 

+ 100 + 99 + 98 . . . . + 3 + 2 + 1 

101 101 101 101 101 101 101 101 101 101 

Sum = 101(100) ÷ 2 = 5,050 

n 

INSTRUCTION 

Help students to learn summation 

notation by translating the 

symbols into English. For example, 

5 

∑ 

n= 1 

3n 

is read as “the sum of 

the values of 3n from n = 1 to 

n = 5.” 

Have students revisit the 

handshake problem which was 

investigated in Lesson 9-1. Ask 

students to write a summation 

notation to represent the number 

of handshakes. 

Make sure that students do not 

assume that the upper limit of a 

summation notation is equivalent 

to the number of terms that are 

being summed. Advise students 

that the index of a summation 

notation is not always 1. 

Make sure that students realize 

that the formula S 

n 

n 

= ( a + a 

n) 

2 1 

does not apply to a summation 

7 

2 

notation such as: ∑ n . 

n= 

1 

Example 3 Reinforce to 

students that the formula 

S 

n 

n 

= ( a + a 

n) is just a 

2 1 

generalization of Euler’s method 

which requires less work. Show 

students how his method could 

be used to determine the number 

of seats in the auditorium. 


INSTRUCTION 

Ongoing Assessment Show 

students how the graph of the 

equation y = 4(x + 1) would 

show all of the terms in this 

series. Emphasize that only the 

points with x-values that are 

natural numbers are part of the 

series. 

WRAP UP 



• Determine if a given sequence 

is arithmetic and state the 

common difference. 

• Use the arithmetic mean to 

determine missing terms in an 

arithmetic sequence. 

• Evaluate a given summation 

notation for an arithmetic 

series. 

Assignment 





pages 422–429 

 

 

 

 

 

 

4( + 1) 

 

= 

1 

 

 

 

 

 

 

 

 

 

 

16 

∑ 

 

 

 

 

 

1 

, 

2 

, 

3 3 1 , 4 , 

5 

,… 1 

3 3 3 

 

 

 

 

 

 

 

 

 

 

 

Think and Discuss Answers 

1. Find the difference between any two consecutive terms of the sequence. 

2. Answers will vary. Sample answer: Use the formula S 

n 

n 

= ( a + a 

n) where 

2 

a 1 

1 

= 1, a n 

= 100, and n = 100. 

3. An arithmetic sequence is a pattern of numbers while an arithmetic series 

is the sum of the numbers in the sequence. 

4. Answers will vary. Sample answer: the arithmetic mean is the average of 

the two numbers. 

5. yes; the sequence 10, 8, 6, 4, 2, … has a common difference of –2. 


50 

∑ 

3 

2( −1) 

 

= 1 

34 

∑3( + 2) 

∑7 

− 5 

= 

1 

53 

∑8 

−1 5 

+ 2 

 

 

= 

1 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

82 

∑ 

= 

1 

62 

= 

1 

25 

∑ 

= 

1 


 

NAME CLASS DATE 

AND SERIES 

RETEACHING 11.2 ARITHMETIC SEQUENCES 

An arithmetic sequence is a sequence in which the difference between consecutive 

terms is the same. The difference between consecutive terms is the common difference 

of the sequence. 

The arithmetic mean of two numbers is the average of the two numbers. The 

arithmetic mean can be used to find a missing term of an arithmetic sequence. 

A series is a mathematical expression for the sum of the terms of a sequence. A series 

can have a finite number of terms that can be counted individually such as 1 + 2 + 3, or 

it can have an infinite number of terms such as 2 + 4 + 6 + 8 + …. An arithmetic series 

is a series whose terms form an arithmetic sequence. 

The sum Sn of an arithmetic sequence with a finite number of terms, such as a1 + a2 + 

a3 + … + an, is given by the formula: Sn = 2 

n (a1 + an), where a1 is the first term, an is 

the last term, and n is the number of terms. 

Summation notation is used to write the terms of a series in a compact way. 

5 

∑ = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 45 

n= 

1 

3n 

Under the Σ summation symbol is the lower limit of the series and an index indicated 

by n = 1. Above the symbol is the upper limit, which is the greatest value of n. To the 

right of the symbol is an explicit formula for the terms of the sequence. 

EXERCISES 

Determine if each sequence is arithmetic. If so, state the common difference of 

the sequence. 

1. 3, 7, 11, 15, 19, … yes; 4 2. 1, 3, 6, 10, 15, 21, … no 

Use the formula Sn = 2 

n (a1 + an) to evaluate each series. 

3. 

12 

∑ 4. 

+ 3n 

2 

−1 

5n 

156 25,150 2n 

1,749 

n= 

1 

100 

∑ 5. ∑ 

n= 

1 

Copyright © CORD 

310 >Algebra 2 Chapter Resource Book 

33 

n= 

1 

 

 

 

 

 

 

 

 

 

 

 


Enrichment 9.2 (CRB) 

 


AND SERIES 

ENRICHMENT 11.2 ARITHMETIC SEQUENCES 


AND SERIES 

EXTRA PRACTICE 11.2 ARITHMETIC SEQUENCES 

Determine if each sequence is arithmetic. If so, state the common difference of the 

sequence. 

1. –1.4, –1.3, –1.2, –1.1, … yes; 0.1 2. 22, 25, 28, 31, 34, … yes; 3 

3. 7, 3, 0, –3, –6, … no 4. 4.2, 5, 5.8, 6.6, 7.4, … yes; 0.8 

5. 30, 24, 18, 12, 6, … yes; –6 6. 0.1, 0.01, 0.001, 0.0001, … no 

7. 1, 2, 4, 8, 16, 32, … no 8. –25, –14, –3, 8, 19, … yes; 11 

Use the arithmetic mean to find the missing term in each arithmetic sequence. 

9. 5, 11, 17, ⁪, 29, … 23 10. 24, ⁪, 16, 12, 8, … 20 

11. 99, 90, ⁪, 72, 63, … 81 12. –0.4, 1.3, 3, ⁪, 6.4, … 4.7 

13. 8, 13, 18, ⁪, 28, … 23 14. 56, ⁪, 24, 8, –8, … 40 

15. 6, ⁪, –8, –15, –22, … –1 16. 4.3, 4.9, ⁪, 6.1, 6.7, … 5.5 

Use the formula Sn = 2 

n (a1 + an) to evaluate each series. 

17. 

20. 

17 

∑ 18. 

n + 6 

8n 

−3 

3n 

255 2,525 315 

n= 

1 

21 

n= 

1 

25 

∑ 19. ∑ 

n= 

1 

n + 3 

n + 3 

n 1 

2 

147 527 –1,225 

∑ 21. 

19 

∑ 24. 

31 

∑ 22. 

n= 

1 


n −5 

23. n − 2 

5n 

+ 10 

314 2 

>Algebra 2 Chapter Resource Book 

152 110.5 1,035 

n= 

1 

26 

n= 

1 

14 

n= 

1 

50 

∑ − + 

n= 

1 

∑ 25. ∑ 

18 

n= 

1 





Applications 






lesson. 

5. a. 500(190 + 7(0)) = 

500(190) = 95,000 

b. 500(190 + 7(6)) = 

500(190 + 42) = 

500(232) = 116,000 

c. Year 2 = 500(190 + 7) = 

500(197) = 98,500 

Year 3 = 500(190 + 14) = 

500(204) = 102,000 

Year 4 = 500(190 + 21) = 

500(211) = 105,500 

Year 5 = 500(190 + 28) = 

500(218) = 109,000 

Year 6 = 500(190 + 35) = 

500(225) = 112,500 

95,000 + 98,500 + 

102,000 + 105,500 + 

109,000 + 112,500 + 

116,000 = 738,500 

738,500 > 735,800, 

so Yoshi can expect to 

produce enough Sheet 

of seaweed over the next 

7 years to be profitable. 

9. a. S n 

= n ( a + a n 

) 

2 

1 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b. S 9 

= 9 10 66 

2 ( + ) = 9(38) 

= 342 

c. 335 = 9 ( 

2 25 + a9 

) 

335 = 112.5 + 4.5a 9 

222.5 = 4.5a 9 

49.4 = a 9 

50 signatures 

d. The funding will increase 

because Alberto gathered 

more signatures. 

402 Chapter 9 Sequences and Series 

11. a. arithmetic sequence 

b. Each trial, total miles decreases 

by 39 so Yvonne’s mileage 

should be 388 – 39 = 299 

miles for a 7th trial. 

c. S n 

= n 533 260 

2 ( + ) 

3,172 = n 2 ( 793) 

3,172 = 396.5n 8 = n 

d. The number of mpg decrease 

by 3 miles in each trial. The 

number of mpg multiplied 

by the number of gallons the 

tank holds gives the total miles 

traveled. 

5 

∑ 

n= 

0 

13( 41− 

3n) 

12. a. 22 – 13 = 9; 40 – 31 = 9 

The common difference is 9. 

b. 4 + 9n 

c. 4 + 9(7) = 67


Vocabulary 

geometric sequence 

common ratio 

geometric mean 

geometric series 




Assignment 



 

 

 

 

 

 

 

 

 

 

 

 

 

 


Applying 

 

 

Investigate with students some real world applications of geometric sequences 

such as compound interest and radioactive decay. Use a spreadsheet to quickly 

generate a large number of terms for the sequences. In the case of compound 

interest, point out to students that the interest rate is the common ratio and 

the opening balance is the first term of the sequence. 


Exercises 8, 10, and 13 from 

pages 422–429 

START UP 

Students have been exposed to 

geometric sequences when they 

studied exponential functions in 

Chapter 9. Make a table of values 

for the function f(x) = 2 x and 

point out the pattern that exists in 

the second column of the table. 

INSTRUCTION 

Point out to students that neither 

the common ratio nor the first 

term for a geometric sequence 

can be zero. Ask students to 

explain the effect that a negative 

common ratio would have on a 

geometric sequence. 

9.3 Geometric Sequences and Series 403

INSTRUCTION 

Tell students that the geometric 

mean of two numbers is the 

number that is equally distant 

between the two numbers in 

multiplication. 

Point out that since there is 

a common ratio between 

consecutive terms in a geometric 

sequence, that the ratio between 

every other term is also a 

constant. The ratio between every 

other term is twice the common 

ratio. 

Example 3 Give students both 

a letter size and a legal size piece 

of paper. Have students measure 

the length and width of the paper 

and compute the ratios. 

 

 

 

 

number number 

 

 

 

 

 

 

 

 

1560 

 

 

 

 

 

 

 

 

 

100 

 

= 

 

210 

2 

= 21, 

000 

144. 

9 

 

 


Show students the following alternative way to 

determine the missing term in Example 2. 

Step 1: The ratio between the terms 15 and 60 is 4. 

Step 2: Because the difference between every other term in an geometric 

sequence is twice the common ratio, the common ratio is 4 ÷ 2 = 2. 

Step 3: Multiplying 15 by the common ratio yields a missing term of 30. 


n 

a( r 

Sn 

= 1 ) 

1 

1 

r 

 

 

 

 

 

 

 

 

 

 

 

 

n 

a1 

( 1 

r ) 

Sn 

 

1 

r 

6 

21 ( 2 ) 

Sn 

 

1 

2 

S 126 

n 

 

 

INSTRUCTION 

Example 4 Have students use 

a graphing calculator to create 

the geometric sequence for this 

example. Make sure to point out 

to students that the graph of this 

sequence forms an exponential 

curve. 

Show students that the 

summation notation for this 

problem could be written as 

6 

∑ 

follows: 2 

n= 1 

n 

Ask students to modify the 

problem so that each person 

calls three additional people. Ask 

students to use the summation 

formula to determine the number 

of people called after the 6th 

branch of the phone tree. 

 

 

 

 


Transferring 

Have students research the Golden Mean, which is a special case of the 

geometric mean of two numbers. Students should provide both a description 

and a formula for calculating a Golden Mean. 


Problem Solving 

Before solving this problem using 

the techniques introduced in this 

section, have students solve the 

problem concerning Roberta’s 

birthday savings account on a 

spreadsheet. Encourage students 

to use the SUM formula in the 

spreadsheet representation of this 

problem. 

Understand the Problem 

You know that Roberta’s 

grandfather put $10 + $20 

+ $40 + … into her savings 

account for 10 years. You need 

to find the sum of the amounts 

deposited into the savings 

account. 

Develop a Plan 

geometric series; S n 

= a r n 

1( 1 − ) ; 

1− 

r 

a 1 

= 10, r = 2, n = 11 

Carry Out the Plan 

S 11 

= 10 1 2 11 

( − ) 

= $20,470 

1− 

2 

17th birthday: S 10 

= 10 1 2 10 

( − ) 

= 

1− 

2 

$10,230 

16th birthday: S 9 

= 10 1 2 9 

( − ) 

= 

$5,110 

1− 

2 

Check the Results 

The amount of money deposited 

on Robert’s 17th birthday is 

$10,230 – $5,110 = $5,120; 

the amount of money deposited 

on Robert’s 18th birthday is $ 

20,470 – $10,230 = $10,240; 

the sum seems reasonable. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. The differences between consecutive terms of a geometric sequence vary, 

but the ratios are the same. In an arithmetic sequence, the differences are 

the same, but the ratios vary. 

2. yes; the geometric sequence 1, –2, 4, –16, 32, … has a common ratio 

of –2. 

3. Divide any term of the sequence by the term that immediately precedes it. 

4. Answers will vary. Sample answer: the amount of money in a CD with 

compounded interest. 

5. Take the square root of the product of the numbers. 


1 1 

2 

, 1 1 

3 

, 1 

4 

, 5 

, 6 

,… 

 

 

 

 

 

WRAP UP 



• Determine if a given sequence 

is geometric and state the 

common ratio. 

• Use the geometric mean to 

determine missing terms in a 

geometric sequence. 


 

 

 

n 

a ( r 

Sn 

1 ) 

1 

1 

r 

 

 

 

 

 

 

 

 

 

 

 

1 

 

1 

 

2 

… 102 3 

10 5 5 

10 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

Assignment 




Exercises 8, 10, and 13 from pages 422–429 



Applications 






lesson. 

8. a. 2(3) 0 = 2(1) = 2 

4 

b. ∑ 23 ( ) n 

= 2(3) 0 + 2(3) 1 

n= 0 

+ 2(3) 2 + 2(3) 3 + 2(3) 4 

= 2 + 6 + 18 + 54 + 162 

= 242 

c. total ratings = 242 + 

= 250 

no; Misty has a 242 

250 = 

96.8% positive feedback 

rating, which is less than 

98%. 

10. a. Convert the times to 

seconds. 

220 242 

= 1.1; 

200 220 = 1.1; 

266. 2 = 1.1; 

292. 

82 

242 266. 

2 

= 1.1; 

322. 

102 

= 1.1 

292. 

82 

The common ratio is 1.1. 

b. 322.102(1.1) = 354.3122 

= 5 min 54.3122 s 

c. 2 h 28 min = 8,880 s 

8,880 + 200 + 220 + 

242 + 266.2 + 292.82 

+ 322.102 + 354.3122 

= 10,777.4342 ≈ 

179.62 min 

Paige’s goal was 3 hours or 

180 minutes. 

179.62 min < 180 min, so 

yes, Paige met her goal. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13. a. 120 48 

300 

= 0.4; 

120 = 0.4; 

19. 

2 

48 

= 0.4; 768 . 

= 0.4; 

19. 

2 

3. 

072 

= 0.4 

768 . 


3.072(0.4) = 1.2288 kb/s 

 

 

 

10 a 

d 

 

2 

 

 

 

 

 

10a 

 

 

25, 

000 

 

( x 500) 

2 

 

( ) 2 

500 

250, 

000 

 

 

 

b. 1.2288(0.4) = 0.49152 kb/s, therefore 400 users would be on the 

network. 


a 

S 

r 

 

 

 

 

 

 


Experiencing 

 

 

805 ( . ) + 1 

= 

1 

 

∞ 

∑ 

 

Have students calculate the area under the curve formed by the parabola 

y = –x 2 + 9 and the x-axis. Instruct students to accomplish this by drawing a 

series of circumscribed rectangles under the curve and computing the area of 

each of these rectangles. Ask students how the size of the rectangle will affect 

the approximation of the area under a curve. 


Vocabulary 

infinite geometric series 

converges 

diverges 

point of discontinuity 

asymptote 




Assignment 




Exercises 4 and 14 from 

pages 422–429 

START UP 

Tell students that convergence 

of an infinite series is the basis 

for the Fundamental Theorem 

of Calculus. Discuss with 

students the fact that while 

both arithmetic and geometric 

sequences can be infinite, only 

geometric series converge. 

INSTRUCTION 

Show students this infinite 

geometric series on a spreadsheet. 

In column A, generate 10 terms. 

In column B, generate 50 terms. 

In column C, generate 100 terms. 

Use the Sum Formula at the end 

of each column. Emphasize the 

convergence of the sums. 

a1 

Use the formula S = 

1− r 

to 

verify that the sum of this series 

diverges to 4 as shown in the 

spreadsheet. 

9.4 Infinite Geometric Series 409

INSTRUCTION 

Students might incorrectly state 

that a geometric series in which 

|r| >1 converges to ∞. Tell 

students that geometric series 

only converge to a real number 

and that ∞ is not considered a 

real number. 

Example 2 Tie a washer onto 

the end of a string to model 

the pendulum discussed in this 

problem. 

Note that in this example, the 

pendulum never really stops. 

However, the series converges. 

Have students discuss why the 

pendulum will eventually stop 

(friction). 

Ongoing Assessment Point 

out to students that while terms 

of the geometric series alternate 

between positive and negative, the 

sum of the terms remains positive 

and converges towards 5 6 . 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a1 

S = 

1 

r 

S = 

48 

1 0. 

99 

S = 4, 

800 

 

 

1 − 1 1 1 

5 

+ 25 

− 125 

+… 5 6 

 

Diversity in the Classroom 

Visual Learner 

Direct students to a website that provides animated demonstrations of fractal 

geometry. Ask students to do additional internet research to compile a list of 

five occurrences of fractal geometry in nature. 


∞ 

3 4 

∑ ( 

=0 

3) 

 

 

 

∞ 

3 1 

4 

+ 4 

∑ ( ) 

 

4 3 = 

0 

9 

 

4 3 

 

 

ACTIVE LEARNING 

Length of Perimeter 

Iteration 

Side of Shape 

1 1 

2 

3 

5 

 

 

4 9 

 

 

2 3 

5 

 

 

 

2 3 

Point out to students how the 

Koch snowflake is beginning to 

approximate a curve, but is still 

constructed of straight lines. 

Students will think that the area 

of the Koch Snowflake will not 

converge since it appears as if 

new area is being added to the 

figure with each iteration. 

To make sure that students 

understand the behavior of the 

Koch snowflake, have them 

construct and fill out a table 

similar to one shown below: 

Area of 

Shape 

 


WRAP UP 



• Determine if an infinite 

geometric series diverges or 

converges. 

• Find the sum of an infinite 

geometric series which 

converges. 

Assignment 





pages 422–429 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∞ 

12 2 

∑ ( 3) 

 

=1 1 1 1 1 

2 

4 

8 

… 

 

 

∞ 

∑051 . ( . 000001) 

= 1 

 

 

∞ 

∑10( 0. 308) 

= 1 

∞ 

01 101 

 

∑ . 

100 

= 

1 

( ) 

 

 

 

 

 

1 3 

∞ 

8 540 000 7 

∑ 

= 

1 

15 

 

 

 

, , ( ) 

 


1. Answers will vary. Sample answer: 1 + 0.5 + 0.25 + 0.125 + … 

converges; 2 + 4 + 8 + 16 + … diverges 

2. Answers will vary. Sample answer: when the absolute value of the 

common ratio is less than 1. 

3. yes; as long as the absolute value of the common ratio is less than 1, 

the series will converge. 

4. If a series converges, it gets closer and closer to a particular sum. If it 

diverges, it does not approach a particular sum. 

5. the first term of the corresponding geometric sequence 


1 − 1 1 1 

2 

+ 4 

− 8 

+… 2 3 1 2 

10 2 3 

∞ 

∑ 

 

∞ 

=1 

( ) 

 

 

 

180( 025 . ) 

∞ 

∑ 

∑08 1 

= 1 

10 

 

4 45 

= 

1 

∞ 

∑ ( . ) 55 5 

= 1 

9 

. ( ) 

500 01 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 2 02 . 02 . 2 

= = 

1− 

0. 

1 09 . 9 

 

 

 

 

2 1 

 

 

 

 

 

4 1 

 

 

 

 

 

 

 

 

 

 


Enrichment 9.4 (CRB) 

 


Mixed Review 

Additional Answers 

35. a. 20, 22, 24, 26, 28 

b. S n 

= n 2 (20 + a ) n 

 

 

 

 

 

 


Applications 






lesson. 

4. a. 9,500 + 5,700 + 3,420 

+ 2,052 + 1,231.2 + 

738.72 + … 

b. a 1 

= 9,500; r = 5 , 700 

9, 

500 

= 0.6 

a 7 

= 9,500(0.6) 6 

= 443.232 

∞ 

∑ 

c. 9, 500( 06 . ) n 

n= 0 

14. a. infinite geometric series 

b. r = 3 1 = 3 

|3| ≥ 1, so the series 

diverge. 

c. Because the series diverges, 

it does not approach a 

particular sum. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

−2 

 

= 

3 

5 + 10 2 

− 4 

5 15 

2 − 2 

= 2 −1 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Remind students of the Fibonacci Sequence which 

was studied earlier in this chapter. Challenge 

students to find how the Fibonacci Sequence 

is represented in Pascal’s Triangle. Give hints as 

necessary. 

 


Vocabulary 

Pascal’s Triangle 

Binomial Theorem 

n factorial 

combinations 




Enrichment 9.5 

Assignment 





pages 422–429 

START UP 

Tell students to rewrite the 

following expressions without 

parentheses. (x + y) 0 , (x + y) 1 , 

(x + y) 2 , and (x + y) 3 . Students 

will often make the error of 

distributing the exponents 

through the parentheses. 

INSTRUCTION 

Ask students to identify all of the 

patterns that they can find in 

Pascal’s Triangle. 

Answer to Activity 

3. 

1 

1 1 

1 2 1 

1 3 3 1 

1 4 6 4 1 

1 5 10 10 5 1 

1 6 15 20 15 6 1 

1 7 21 35 35 21 7 1 

1 8 28 56 70 56 28 8 1 

9.5 The Binomial Theorem 415

INSTRUCTION 

Ask students to express why the 

factorial function is recursive in 

nature. 

Evaluate the combinations 4 

C 0 

, 

C , C , C , and C . Ask students 

4 1 4 2 4 3 4 4 

to identify the pattern in the 

numbers as well as the matching 

row in Pascal’s Triangle. 

Students often ask about the 

factorial of 0 and the factorial 

of a negative number. Tell 

students that the factorial of 0 

is one (much in the same way as 

a 0 = 1) and that the factorial of a 

negative number is undefined. 

Tell students that combinations 

will be studied in more depth in 

Chapter 14 and that for right now 

the given formula will suffice. 


 


RETEACHING 11.5 THE BINOMIAL THEOREM 

Binomials can be multiplied using the FOIL method and the distributive property, but 

Pascal’s Triangle can also be used to expand a binomial that is raised to a single power. 

Pascal’s Triangle is the triangular array of numbers shown below. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n! 

 

r!( n 

r)! 

1 

1 1 1 1 1 

2 1 1 

3 3 4 1 

6 5 10 4 1 

10 5 1 

The entries of the rows of Pascal’s Triangle match the coefficients of a binomial 

expansion. 

Row 1: (a + b) 0 = 1 

Row 2: (a + b) 1 = 1a + 1b 

Row 3: (a + b) 2 = 1a 2 + 2ab + 1b 2 

Row 4: (a + b) 3 = 1a 3 + 3a 2 b + 3ab 2 + 1b 3 

In each expansion of (a + b) n , the exponents of a begin with n and decrease to 0. The 

exponents of b begin with 0 and increase to n. The coefficients of the terms are the 

entries of row n + 1 of Pascal’s Triangle. 

The Binomial Theorem can be used as a general rule for expanding a binomial. 

The expression n!, read n factorial, is defined for any positive integer n to be the 

following: n! = n · (n – 1) ּ … ּ 3 ּ 2 ּ 1. For n = 0, n! = 1. 

The Binomial Theorem states that for every positive integer n: 

(a + b) n = nC0a n + nC1a n – 1 b + nC2a n – 2 b 2 + … + nCn – 1ab n – 1 + nCnb n . 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the definition of the Binomial Theorem, the coefficients nCr are combinations and 

are defined as nCr = 

EXERCISES 

n! 

r!( n− 

r)! 

, for 0 ≤ r ≤ n. 

Use Pascal’s Triangle to expand each binomial. 

1. (c + d) 2 2. (s + t) 3 

c 2 + 2cd + d 2 s 3 + 3s 2 t + 3st 2 + t 3 

3. (p – q) 5 4. (2m + n) 4 

p 5 – 5p 4 q + 10p 3 q 2 – 10p 2 q 3 + 5pq 4 – q 5 16m 4 + 32m 3 n + 24m 2 n 2 + 8mn 3 + n 4 




Experiencing 

Show students how to use a 

graphing calculator to compute 

factorials and combinations. 


INSTRUCTION 

Provide students with a piece of 

graph paper to construct Pascal’s 

Triangle. 

Point out the symmetry in Pascal’s 

Triangle. Show students that if 

they were to fold the triangle 

along its altitude, that the 

numbers would match up. 

Tell students that although 

Pascal’s Triangle can be traced 

back to China in the 12th century, 

it was named for the French 

Mathematician Blaise Pascal in the 

17th century. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Substitute 3c for a, –4d for b, and 4 for n in the Binomial Theorem: 

(3c – 4d) 4 = 4 

C 0 

(3c) 4 + 4 

C 1 

(3c) 3 (–4d) + 4 

C 2 

(3c) 2 (–4d) 2 + 4 

C 3 

(3c)(–4d) 3 + 

4 C 4 (–4d)4 = 81c 4 – 432c 3 d + 864c 2 d 2 – 768cd 3 + 256d 4 . 

3. Substitute 2p for a, 5q for b, and 6 for n in the general form of the 3rd 

term of the binomial expansion: 6 

C 2 

(2p) 4 (5q) 2 = 400p 4 q 2 . 

4. The expression 6 

C 3 

means “6 items chosen 3 at a time.”; it is evaluated 

as 

6! 

. 

3!( 6− 

3)! 

5. Multiply n by each of the positive integers less than it until you reach the 

integer 1. 

Think and Discuss 

Answers 

1. The coefficients of the terms 

of the expansion are the 

entries of a row of Pascal’s 

Triangle. 


WRAP UP 



• Use Pascal’s Triangle to expand 

a binomial. 

• Use the Binomial Theorem to 

expand a binomial. 

• Use the Binomial Theorem 

to find a specified term of a 

binomial expansion. 

Assignment 





pages 422–429 

Practice and Problem 

Solving Additional 

Answers 

6. a 2 + 2ab + b 2 

7. r 2 – 2rt + t 2 

8. m 3 + 3m 2 n + 3mn 2 + n 3 

9. c 4 + 4c 3 d + 6c 2 d 2 + 4cd 3 

+ d 4 

10. x 5 – 5x 4 y + 10x 3 y 2 – 10x 2 y 3 

+ 5xy 4 – y 5 

11. p 4 + 4p 3 + 6p 2 + 4p + 1 

12. q 3 – 9q 2 + 27q – 27 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13. r 6 – 12r 5 + 60r 4 – 160r 3 + 19. x 5 – 10x 4 y + 40x 3 y 2 – 80x 2 y 3 + 80xy 4 – 32y 5 

240r 2 – 192r + 64 

20. x 6 – 6x 5 + 15x 4 – 20x 3 + 15x 2 – 6x + 1 

14. h 2 – 2hk + k 2 

21. a 6 – 24a 5 + 240a 4 – 1,280a 3 + 3,840a 2 – 6,144a + 4,096 

15. j 3 + 3j 2 k + 3jk 2 + k 3 

16. x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 

17. m 5 + 5m 4 + 10m 3 + 10m 2 + 

5m + 1 

18. c 4 + 12c 3 d + 54c 2 d 2 + 

108cd 3 + 81d 4 


2 3 1 3 

 

 

 

 

 

 

 

 

 

 


Applications 






lesson. 

1. a. 28 – 21 = 7 

28 + 7 + 1 = 36 

b. Calculate the difference 

between the last two terms 

and then add 1. Add the 

result to the last term in 

the sequence to find the 

next term in the sequence. 

c. n 3 

= (n 2 

– n 1 

+ 1) + n 2 

= 

2n 2 

– n 1 

+ 1 

2. a. 2 0 , 2 1 , 2 2 , and 2 3 

b. Let the row number 

equal n. 

n – 1 = 8 

n = 9 or 9th row 

c. 2 8 = 1 + 8 + 28 + 56 

+ 70 + 56 + 28 + 8 + 1 

= 256 


 


EXTRA PRACTICE 11.5 THE BINOMIAL THEOREM 

Use Pascal’s Triangle to expand each binomial. 

1. (y – z) 2 2. (m + n) 4 

y 2 – 2yz + z 2 m 4 + 4m 3 n + 6m 2 n 2 + 4mn 3 + n 4 

3. (a – b) 3 4. (x + y) 5 

a 3 – 3a 2 b + 3ab 2 – b 3 x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5 

5. (c – d) 5 6. (w + 1) 4 

c 5 – 5c 4 d + 10c 3 d 2 – 10c 2 d 3 + 5cd 4 – d 5 w 4 + 4w 3 + 6w 2 + 4w + 1 

7. (m – 2) 6 m 6 – 12m 5 + 60m 4 – 160m 3 + 240m 2 – 192m + 64 

Use the Binomial Theorem to expand each binomial. 

8. (x + y) 4 9. (a + 2b) 4 

x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 a 4 + 8a 3 b + 24a 2 b 2 + 32ab 3 + 16b 4 

10. (a + 3b) 4 11. (n – 1) 5 

a 4 + 12a 3 b + 54a 2 b 2 + 108ab 3 + 81b 4 n 5 – 5n 4 + 10n 3 – 10n 2 + 5nd 4 – 1 

12. (t – 1) 6 13. (3a + b) 4 

t 6 – 6t 5 + 15t 4 – 20t 3 + 15t 2 – 6t + 1 81a 4 + 108a 3 b + 54a 2 b 2 + 12ab 3 + b 4 

14. (x – 3y) 5 x 5 – 15x 4 y + 90x 3 y 2 – 270x 2 y 3 + 405xy 4 – 243y 5 

Find the specified term of each binomial expansion. 

15. the fourth term of (x + y) 8 16. the seventh term of (a + b) 11 

56x 5 y 3 462a 5 b 6 

17. the fifth term of (c – d) 9 18. the fourth term of (m – 2n) 6 

126c 5 d 4 –160m 3 n 3 

19. the third term of (2y – 5z) 5 20. the eighth term of (a + b) 20 

2,000y 3 z 2 77,520a 13 b 7 




MATH LAB 

Activity 1 

PREPARE 

• An open spread of newspaper 

will work best for this lab. 

Newspaper works best because 

it is thin and large enough to 

be folded many times. 

TEACH 

• This lab should be completed 

individually. 

• You can select a couple 

of students to follow the 

steps in the lab, but fold 

the newspaper in half using 

the direction that the rest 

of the class does not use 

(folding vertically instead of 

horizontally). This will allow 

for additional discussion in the 

follow up of the activity. 

FOLLOW-UP 

• Have a class discuss about how 

many layers are involved in 

Steps 9 and 10. Ask students 

to think about real-world 

applications that model 

geometric sequences, such 

as this. Ask students to name 

the type of function involved. 

(exponential) 

• If you had a students fold their 

newspapers differently that the 

rest of the class, discuss if they 

were able to fold the same 

number of times as everyone 

else. If they were able to 

fold more or less, discuss the 

implications this would have 

on other applications such as 

bacteria growth, or compound 

interest. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 5 

 

Math Lab Notes and Solutions 

 

 

 

 

 

7. geometric sequence; the ratio between consecutive terms in the sequence 

is always 2 


70 = ( + ) 

 

 

 

 

2 15 5 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Math Lab Notes and Solutions 

 

2. n = 7; the pyramid cannot be built because the common difference is not 

a whole number and therefore cannot be represented by a penny. 

3. 70 = n 2 (20 + 8); n = 5; the common difference is 3. The arithmetic 

sequence is 20, 17, 14, 11, 8. 

4. 90 = n 2 

(18 + 12); n = 6; the pyramid cannot be built because the 

common difference is not a whole number and therefore cannot be 

represented by a penny. 

MATH LAB 

Activity 2 

PREPARE 

• This lab is interactive and 

intuitive. Be sure that each 

group has a roll of pennies. 

TEACH 

• Students should work in 

groups of 3 students. 

• Roles for students are as 

follows: 

1. recorder 

2. pyramid builder 

3. calculator 

• Students will have to build the 

pyramids and test for common 

differences in Steps 2, 3, 

and 4. 

FOLLOW-UP 

• As a class, discuss the 

applications of this type of 

situation. If anyone in the class 

has worked in a store and had 

experience building displays, 

invite them to share their 

experiences. 

• Discuss the advantages to 

calculating before you start 

to build and the results in you 

begin building before you 

know the number of rows, the 

starting number in the row, 

the ending number in the row, 

and the common difference. 

Math Labs 421


Solutions and Notes 

1. a. 28 – 21 = 7 

28 + 7 + 1 = 36 

b. Calculate the difference 

between the last two terms 

and then add 1. Add the 

result to the last term in 

the sequence to find the 

next term in the sequence. 

c. n 3 

= (n 2 

– n 1 

+ 1) + n 2 

= 

2n 2 

– n 1 

+ 1 

2. a. 2 0 , 2 1 , 2 2 , and 2 3 

b. Let the row number 

equal n. 

n – 1 = 8 

n = 9 or 9th row 

c. 2 8 = 1 + 8 + 28 + 56 

+ 70 + 56 + 28 + 8 + 1 

= 256 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 




3. a. 125 + 0.4x 

b. 125 + 0.4(300) = 125 + 

120 = 245 

c. 25 + 0.6x 

d. 25 + 0.6(200) = 25 + 

120 = 145 

4. a. 9,500 + 5,700 + 3,420 + 

2,052 + 1,231.2 + 738.72 

+ … 

b. a 1 

= 9,500; r = 5 , 700 

9, 

500 

= 0.6 

a 7 

= 9,500(0.6) 6 = 

443.232 

∞ 

∑ 

c. 9, 500( 06 . ) n 

n= 0 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∞ 

∑ 9, 500( 06 . ) 

= 0 

 

Math Applications 423



5. a. 500(190 + 7(0)) = 

500(190) = 95,000 

b. 500(190 + 7(6)) = 

500(190 + 42) = 

500(232) = 116,000 

c. Year 2 = 500(190 + 7) 

= 500(197) = 98,500 

Year 3 = 500(190 + 14) 

= 500(204) = 102,000 

Year 4 = 500(190 + 21) 

= 500(211) = 105,500 

Year 5 = 500(190 + 28) 

= 500(218) = 109,000 

Year 6 = 500(190 + 35) 

= 500(225) = 112,500 

95,000 + 98,500 + 

102,000 + 105,500 + 

109,000 + 112,500 + 

116,000 = 738,500 

738,500 > 735,800, 

so Yoshi can expect to 

produce enough Sheet of 

seaweed over the next 7 

years to be profitable. 

6. a. a 2 

= $750,000(1.1) 

= $825,000 

b. a 3 

= $825,000(1.1) 

= $907,500 

$905,000 < $907,500, so 

Mari did not earn a bonus. 

c. a 5 

= $750,000(1.1) 5–1 

= $750,000(1.4641) 

= 1,098,075 

d. yes; a 2 

would now equal 

$900,000 instead of 

$825,000 and would 

increase each subsequent 

number. 

e. a n 

= 1.1(a n–1 

) 

 

 

 

 

 

 

 

6 

∑ 

500( 190 + 7) 

 

 

 

 

= 

0 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


4 

23 ( ) 

 

∑ 

= 0 

 

 

 

 



b. 0.68(9) = $6.12 

8. a. 2(3) 0 = 2(1) = 2 

4 

b. 23 ( ) n 

n= 0 

250 

 

242 

 

7. a. For each 1 cent change 

in earnings per share, the 

closing share price changes 

0.68. The expression is 

0.68x. 

∑ = 2(3) 0 + 2(3) 1 + 

2(3) 2 + 2(3) 3 + 2(3) 4 = 2 

+ 6 + 18 + 54 + 162 = 

242 

c. total ratings = 242 + = 

250 

no; Misty has a 242 

250 = 

96.8% positive feedback 

rating, which is less than 

98%. 




9. a. S n 

= n ( a + a n 

) 

2 

1 

b. S 9 

= 9 10 66 

2 ( + ) = 9(38) 

= 342 

c. 335 = 9 ( 

2 25 + a 

9) 

335 = 112.5 + 4.5a 9 

222.5 = 4.5a 9 

49.4 = a 9 

50 signatures 

d. The funding will increase 

because Alberto gathered 

more signatures. 

10. a. Convert the times to 

seconds. 

220 242 

= 1.1; 

200 220 = 1.1; 

266. 2 = 1.1; 

292. 

82 

= 

242 266. 

2 

1.1; 

322. 

102 

= 1.1 

292. 

82 


b. 322.102(1.1) = 354.3122 

= 5 min 54.3122 s 

c. 2 h 28 min = 8,880 s 

8,880 + 200 + 220 + 

242 + 266.2 + 292.82 

+ 322.102 + 354.3122 

= 10,777.4342 ≈ 

179.62 min 

Paige’s goal was 3 hours or 

180 minutes. 

179.62 min < 180 min, so 

yes, Paige met her goal. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 




11. a. arithmetic sequence 

b. Each trial, total miles 

decreases by 39 so 

Yvonne’s mileage should 

be 388 – 39 = 299 miles 

for a 7th trial. 

c. S n 

= n 533 260 

2 ( + ) 

3,172 = n 2 ( 793) 

3,172 = 396.5n 

8 = n 

d. The number of mpg 

decrease by 3 miles in 

each trial. The number 

of mpg multiplied by the 

number of gallons the tank 

holds gives the total miles 

traveled. 

5 

∑ 

n= 

0 

13( 41− 

3n) 

 

 

 

 

 

5 

∑13( 41− 

3) 

= 

0 

 




12. a. 22 – 13 = 9; 40 – 31 = 9 

The common difference 

is 9. 

b. 4 + 9n 

c. 4 + 9(7) = 67 

13. a. 120 48 

300 

= 0.4; 

120 = 0.4; 

19. 

2 

48 

= 0.4; 768 . 

= 0.4; 

19. 

2 

3. 

072 

= 0.4 

768 . 


3.072(0.4) = 1.2288 kb/s 

b. 1.2288(0.4) = 0.49152 

kb/s, therefore 400 users 

would be on the network. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 




14. a. infinite geometric series 

b. r = 3 1 = 3 

|3| ≥ 1, so the series 

diverge. 

c. Because the series diverges, 

it does not approach a 

particular sum. 

15. a. Each term is the product of 

the previous two terms, so 

a n 

= a n–2 

• a n–1 

b. 0.3 = 0.5(a n–2 

) 

0.6 = a n–2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Vocabulary Review 

arithmetic mean (9.2) 

arithmetic sequence (9.2) 

asymptote (9.4) 

Binomial Theorem (9.5) 

combinations (9.5) 

common difference (9.2) 

common ratio (9.3) 

converges (9.4) 

diverges (9.4) 

explicit formulas (9.1) 

Fibonacci sequence (9.1) 

geometric mean (9.3) 

geometric sequence (9.3) 

geometric series (9.3) 

infinite geometric series (9.4) 

Pascal’s Triangle (9.5) 

point of discontinuity (9.4) 

n factorial (9.5) 

recursive formula (9.1) 

sequence (9.1) 

series (9.2) 

summation notation (9.2) 

term (9.1) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14 2 

 

 

 

 

 

 

 

 

 

 

 

1 1 

4 

, 1 1 

5 

, 1 1 

6 

, 7 

, 8 

, 9 

,… 

 

a 

1 

3 ; 1 

= n 

n + 

 

 

 

 

 

 

 

 

50 

2( + 3) 

 

∑ 

= 

1 

12 

∑ 

2 

− 3+ 

7 

= 

1 

 

 

 

 

 

 

 

 

 

18 

 


1 

8 , 1 4 , 1 2 ,1,2,4, 

 

1 2 1 2 1 4 1 4 1 8 

 

n 

a1 

( 1− 

r ) 

 

Sn 

= 

1− 

r 

1 10 

( − 

Sn 

= 8 1 2 ) 

1− 

2 

S = 127.875 

 

 

 

 

 

∞ 

15 1 

∑ ( 3 ) 

=1 1 3 

a1 

1 3 

S = 

1 − r 

S = 5 

1− 

 

1 3 

 

S = 75 . 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∞ 

 

120 08 . 

∑ 

= 

1 

( ) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chapter Review 

Additional Answers 

Lesson 9.4 

10. diverges 

11. converges; 480 

12. 113 1 3 m 

 

Chapter Review 431

1 2 10 , 922 1 2 

 

 

 

 

∞ 

 

1 

∑ (. 

20 1 001 ) 

= 1 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


1 2 

5 

, 4 

5 

, 5 , 13 5 

,… 

 

 

 

1 

 

1 

2 

4 

 

 

 

 

 

 

 

 

 

 

 

 

2 

+ 3 

+ 2 

3 −22 

−13 

−10 

 

 

 

 

 

∞ 

24( 025 . ) 

∑ 

= 1 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 

5 

− 2 

∑ 

= 

1 

 

 

 

 

 

 

 

 

 

 

4 9 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Standardized Test 

Practice Additional 

Answers 

Open Ended Response 

8. Answers will vary. Sample 

answer: the series 1 + 

0.1 + 0.01 + 0.001 + … 

converges, and the series 2 + 

4 + 8 + 16 + … diverges. 

9. 04 . 

= 

04 . 

= 

4 

101 . 009 . 9 

Standardized Test 

Response Form (CRB) 

 

Chapter Assessments 433

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